{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2024:5PKDYYDBRAI6XSHQ6V7IRPPLBJ","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"d553401c152d31cdcdd2f2d4dc57e9c06b417289f9739335aaf30f24856d6792","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.AP","submitted_at":"2024-06-22T11:44:41Z","title_canon_sha256":"b35688aa2c718df3dfe69dddd4d42b480e8058b2fa7069ad9634fff7e74b6635"},"schema_version":"1.0","source":{"id":"2407.01588","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2407.01588","created_at":"2026-07-05T08:39:02Z"},{"alias_kind":"arxiv_version","alias_value":"2407.01588v1","created_at":"2026-07-05T08:39:02Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2407.01588","created_at":"2026-07-05T08:39:02Z"},{"alias_kind":"pith_short_12","alias_value":"5PKDYYDBRAI6","created_at":"2026-07-05T08:39:02Z"},{"alias_kind":"pith_short_16","alias_value":"5PKDYYDBRAI6XSHQ","created_at":"2026-07-05T08:39:02Z"},{"alias_kind":"pith_short_8","alias_value":"5PKDYYDB","created_at":"2026-07-05T08:39:02Z"}],"graph_snapshots":[{"event_id":"sha256:5344673ef47f819c5486311512fc72f26ba349e6a1c3ccd3a7e720d9bafcc6b7","target":"graph","created_at":"2026-07-05T08:39:02Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/2407.01588/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"In this paper, we study the well-posedness theory and the scattering asymptotics for the energy-critical, Schr\\\"odinger equation with indefinite potential \\begin{equation*}\n  \\left\\{\\begin{array}{l} i \\partial_t u+\\Delta u-V(x)u +|u|^{\\frac{4}{N-2}}u=0,\\ (x, t) \\in \\mathbb{R}^N \\times \\mathbb{R}, \\\\ \\left.u\\right|_{t=0}=u_0 \\in H ^1(\\mathbb{R}^N), \\end{array}\\right. \\end{equation*} where $V(x):\\mathbb{R}^N\\rightarrow \\mathbb{R}$ is indefinite and satisfies appropriate conditions. Using contraction mapping method and concentration compactness argument, we obtain the well-posedness theory in pro","authors_text":"Jun Wang, Zhaoyang Yin","cross_cats":[],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.AP","submitted_at":"2024-06-22T11:44:41Z","title":"Global well-posedness, scattering and blow-up for the energy-critical, Schr\\\"odinger equation with indefinite potential in the radial case"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2407.01588","kind":"arxiv","version":1},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:6e8c800eb3ed59b1ae174f9300bb40119b9b5a59761b486c92062a9e63478ed1","target":"record","created_at":"2026-07-05T08:39:02Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"d553401c152d31cdcdd2f2d4dc57e9c06b417289f9739335aaf30f24856d6792","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.AP","submitted_at":"2024-06-22T11:44:41Z","title_canon_sha256":"b35688aa2c718df3dfe69dddd4d42b480e8058b2fa7069ad9634fff7e74b6635"},"schema_version":"1.0","source":{"id":"2407.01588","kind":"arxiv","version":1}},"canonical_sha256":"ebd43c60618811ebc8f0f57e88bdeb0a4033993da6e4fc3fa83914f535776f48","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"ebd43c60618811ebc8f0f57e88bdeb0a4033993da6e4fc3fa83914f535776f48","first_computed_at":"2026-07-05T08:39:02.465518Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T08:39:02.465518Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"83+slmJefz/Z8pHlGuiBM9tlPTlRyfI07kS5TnplW1Hfy9MKfXdqyUbl2rbBoX+ljgcJcUc2oXgMBRRt2Qe8AA==","signature_status":"signed_v1","signed_at":"2026-07-05T08:39:02.465983Z","signed_message":"canonical_sha256_bytes"},"source_id":"2407.01588","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:6e8c800eb3ed59b1ae174f9300bb40119b9b5a59761b486c92062a9e63478ed1","sha256:5344673ef47f819c5486311512fc72f26ba349e6a1c3ccd3a7e720d9bafcc6b7"],"state_sha256":"83adceda970b9ad5bfca7e13d0d40e99504deddeee6902b2eaead742748c836b"}